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Precise definition of linear combination

  1. Nov 15, 2008 #1
    i know that a linear combination of the vectors {v1,v2,...,vn} is any sum with terms that are scalar multiples of those vectors. But is a1v1 + a2v2 the same linear combination as a2v2 + a1v1? i know they evaluate to the same thing because vector addition is commutative but if i wanted to be precise would i say that it's the same linear combination or a different one only with the same set of coefficients? because if at least one of b1 and b2 was different from a1 and a2, then b1v1 + b2v2 is not considered to be the same linear combination even though they might be equal.
  2. jcsd
  3. Nov 16, 2008 #2


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    Hi redjoker! :smile:

    And stop worrying :wink: … there really isn't a problem! :biggrin:
  4. Nov 16, 2008 #3


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    The point is that linear combination refers to a relation between a vector, say x and a set of vectors say {xi}
    A good definition is of Paul Halmos (in Finite-dimensional vector spaces):
    In other words, the phrase "x is the linear combination of..." is the synonym of "x is linearly dependent on...".
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